Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866912080564584448 |
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| author | Lin, Xun Zhang, Shizhuo |
| author_facet | Lin, Xun Zhang, Shizhuo |
| contents | Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{Jac}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset\mathbb{P}^n(n\geq 2)$ of degree $d$ if $\mathrm{gcd}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22, Theorem 1.2]$ using a similar idea. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09927 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces Lin, Xun Zhang, Shizhuo Algebraic Geometry Mathematical Physics 14F05, 14J45, 14D20, 14D23 Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{Jac}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset\mathbb{P}^n(n\geq 2)$ of degree $d$ if $\mathrm{gcd}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22, Theorem 1.2]$ using a similar idea. |
| title | Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces |
| topic | Algebraic Geometry Mathematical Physics 14F05, 14J45, 14D20, 14D23 |
| url | https://arxiv.org/abs/2310.09927 |